Short-time Fourier transform

Original address: http://blog.sina.com.cn/s/blog_82a927880102uwi1.html

Wavelet and short time to learn together, as for reasons to look down to know.

The basic idea of short-time Fourier transform is to add the signal to the time window, and make Fourier transform the signal in the window, and get the variable frequency spectrum of the signal. Thus its time resolution and frequency resolution are constrained by the Heisenberg uncertainty principle, and once the window function is selected, the time-frequency resolution is determined. This makes it limited to the analysis of mutation signal and non-stationary signal, so it is not a dynamic analysis method, it can not reflect the mutation of signal sensitively, it is only applicable to the analysis of the slow-change signal.
Wigner a ville distribution is defined as the Fourier transform of the Signal center covariance function, which has many excellent properties, such as symmetry, time-shift, combination, complex-conjugate, etc., without loss of amplitude and phase information of the signal, and clear concept of instantaneous frequency and group delay.
Its insufficiency is not guaranteed nonnegative, especially to the multi-component signal or has the complex modulation law signal to produce the serious cross-item interference, this is the two time frequency distribution The intrinsic result, the large number of crosses can drown or seriously disturb the signal's self-item, the fuzzy signal original characteristic. The subsequent modification of the kernel function in the Cohen class puts forward a variety of new time-frequency distributions, such as pseudo-Winger-ville distribution, modified smoothing pseudo-Winger-ville distribution and so on, which plays a great role in suppressing cross-interference, but does not contain cross-term interference and has winger- The time-frequency distribution of ville distributed aggregation is nonexistent.
The wavelet transform can decompose the signal in multi-scale through the operation of scaling and moving, and it is able to obtain various time-frequency information from the signal effectively, it has good localization property in time and frequency domain, and has multi-resolution analysis characteristic.
However, the results of wavelet decomposition rely on wavelet basis function, and the application range of small base function is very inconsistent, which results in the selection of wavelet bases, if the mother wavelet selection is not appropriate, the application effect will be greatly affected; Wavelet analysis does not have self-adaptability, and once it is used to analyze data with multi-frequency components, The result can only reflect the signal in a fixed frequency band, so we should choose different wavelet bases.

 

Gossip article:

1. Fourier Transform defect----ft localization characteristic analysis

FT's outstanding contribution to smooth signal analysis and processing is based on the fact that people use it to transform complex time signals and spatial signals into frequency domains, and then analyze and represent the characteristics of time domain signals with spectral characteristics.

The FT transformation tells us that extracting spectral information from a time (analog) signal is the use of (-∞,∞) time information to calculate the spectrum of a single frequency (the value of any part of the frequency domain process () fω), which is determined by the time domain process on (-∞,∞). Therefore, the Fourier transform to each other's characterization is "global", can not reflect the characteristics of their local areas, although people from the Fourier transform can clearly see a message contains the number of each frequency, but it is difficult to see the different signal launch time and the duration of the launch, Lack of time information makes Fourier analysis fragile and prone to error.

Illinois University Professor Y. I y E R once said: "If you record 1 hours of information and make a mistake in the last 5 minutes, this error will ruin the entire Fourier transform." The phase error is catastrophic, and if you make a mistake in the phase, you will eventually find that what you are doing has nothing to do with the initial signal. ”

In fact, for common unstable signals, such as voice signal, music signal, pulse signal of nuclear detection, and image signal of nuclear medicine, their frequency domain characteristic is changing with time, people need to know what the main frequency characteristic is in some local time period, and also need to know the time period of information of certain frequency. That is, the need to understand the time-frequency localization requirements. For this time-frequency localization requirement, Fourier transform is powerless.

Solutions: Short-time Fourier transform, wavelet transform. Each has its merits and demerits, this text chat point short-time Fourier transform, finally will lead to a little wavelet transform.

2. Short-time Fourier transform (Windowed Fourier transform)

Basic idea: The non-stationary process is regarded as a series of short-term stationary signal superposition, short-time can be achieved by adding windows in time. By this method, one can at least say that whatever happens, it must happen in a particular part of the signal.

In Fourier integration, the time window function is multiplied with the signal to realize the opening and moving of the window near U and then Fourier transform. In linear space There is a measurable, square integrable function, which is a windowed Fourier transform:

Inverse Transformation of window Fourier transform:

3. Wavelet derivation

The analysis of the function (signal) by the Fourier transform of the window is equivalent to a "magnifying glass" with the same shape, size and magnification as the time-frequency phase plane to observe the frequency characteristic of a fixed length. The problem here is that although the windowed Fourier transform solves the localization problem of the transformation function, its window size and shape are fixed, i.e. the window is not self-adaptive. What this means.

But in the actual problem, for the high spectrum information, because the waveform is relatively narrow, the time interval should be relatively small, in order to give a better precision, and further determine the peak and breakpoint, or the need to use a narrow window to reflect the high-frequency components of the information, and for the low spectral information, because the waveform is relatively wide, The time period should be relatively wide to give the complete signal information, or must use a wide time domain window to reflect the low-frequency components of information. With a short-time Fourier transform, if you choose a wide window, the low-frequency components can be seen clearly, in the high-frequency part of the determination of time is very bad; if you choose a narrow window, the high frequency can be very good to determine the time, but at low frequencies may not be loaded.

In this way, it is really appropriate for the "magnifying glass" the length of the width can be changed, it is to achieve such a goal, people introduced the wavelet transform.

Practice article:

There are two main programs, respectively: short-time Fourier transform, inverse short-time Fourier transform

Short-time Fourier Transform

1:function [Stft, F, t] = Stft (x, Wlen, H, Nfft, FS)

2:if size (x,2) > 1

3:x = X ';

4:end

5:xlen = length (x);

6:win = Hamming (Wlen, ' periodic ');

7:rown = Ceil ((1+NFFT)/2);

8:coln = 1+fix ((xlen-wlen)/h);

9:stft = Zeros (Rown, Coln);

10:indx = 0;

11:col = 1;

:% perform STFT

13:while indx + wlen <= Xlen

14:XW = x (Indx+1:indx+wlen). *win;

15:x = FFT (XW, NFFT);

16:stft (:, col) = X (1: (Rown));

17:indx = indx + H;

18:col = col + 1;

19:end

20:t = (wlen/2:h:xlen-wlen/2-1)/fs;

21:f = (0:rown-1) *fs/nfft;

22:end

Inverse Short-time Fourier Transform

1:function [x, T] = Istft (Stft, H, Nfft, FS)

2:coln = Size (Stft, 2);

3:xlen = Nfft + (coln-1) *h;

4:x = Zeros (1, Xlen);

5:win = Hamming (nfft, ' periodic ');

6:if REM (NFFT, 2)

7:for B = 0:h: (h* (coln-1))

8:x = Stft (:, 1 + b/h);

9:X = [x; conj (x (End:-1:2))];

10:xprim = Real (Ifft (X));

11:x ((b+1):(b+nfft)) = X ((b+1):(B+nfft)) + (Xprim.*win) ';

12:end

13:else

14:for B = 0:h: (h* (coln-1))

15:x = Stft (:, 1+b/h);

16:X = [x; conj (x (End-1:-1:2))]; 

17:xprim = Real (Ifft (X));

18:x ((b+1):(b+nfft)) = X ((b+1):(B+nfft)) + (Xprim.*win) ';

19:end

20:end

21:w0 = SUM (win.^2);

22:x = x.*h/w0;

23:actxlen = length (x);

24:t = (0:actxlen-1)/fs;

25:end

Test signal 1:

First, the simple sine signal is chosen to do the test.

Parameters:

1:fs = 48000;

2:t = 0:1/fs:1-1/fs;

3:x = 10*sin (2*pi*t*10);

Test Signal 2:

The choice here is the sound that we used to have when Windows was turned on, very familiar with it.